In principle, a dual-slope circuit doesn't need a reset switch if it performs integration and deintegration phase alternatingly. Deintegration would stop when the integrator output voltage is exactly zero.
Practical dual-slope circuits implement an additional reset or autozero phase after deintegration. In this phase, a reset switch or equivalent reset circuit removes residual integrator charge and provides clean initial conditions for the succeeding integrator phase.
It seems to me that you are more guessing about dual-slope operation than clearly understanding the principle. You may get more clarity by studying existing dual-slope ADC implementations, e.g. ICs like Intersil ICL7106 or Microchip TC500.